The realization problem for J{\o}rgensen numbers

TL;DR

This paper investigates the Jørgensen number for non-elementary Kleinian groups, proving that any value greater than or equal to 1 can be realized, and provides computational visualizations of these numbers.

Contribution

It demonstrates that for any r >= 1, there exists a non-elementary Kleinian group with Jørgensen number r, answering a previously posed question.

Findings

Existence of Kleinian groups with any Jørgensen number >= 1

Computational estimates of Jørgensen numbers in Schottky space

Visualization of Jørgensen numbers through computer-generated images

Abstract

Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space.

Figures

Equations

$$J(G)=\inf \{|{\mathrm{tr}}^{2}A-4|+|\mathrm{tr}[A,B]-2|;G=⟨A,B⟩\}.$$

$$|{\mathrm{tr}}^{2}X-4|+|\mathrm{tr}[X,Y]-2|\ge 1,$$

$$J(X,Y):=|{\mathrm{tr}}^{2}X-4|+|\mathrm{tr}[X,Y]-2|$$

$$J(G):=\inf \{J(X,Y)|⟨X,Y⟩=G\}.$$

$$M([z,z^{\prime }]):=\frac{i}{z^{\prime }-z}\left(\begin{array}{cc}z+z^{\prime }& -2z{z}^{\prime }\\ 2& -z-z^{\prime }\end{array}\right).$$

$$P_a:=M([a,-3a]),Q:=M([1,-1]),R:=M([0,\infty ]).$$

$$⟨P_a,Q,R|P_a^2=Q^2=R^2=(QR{)}^2=(R{P}_a{)}^3=id⟩$$

$$\mathrm{tr}{E}_a\in \{\pm x|x\in \mathbb{R},x\ge 3\}\cup \{\pm yi|y\in \mathbb{R},y\ge 1\}.$$

$$M=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ 0& \sqrt{2}\end{array}\right).$$

$$M{P}_1{M}^{-1}=\left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right),MQ{M}^{-1}=\left(\begin{array}{cc}-i& 0\\ -2i& i\end{array}\right),MR{M}^{-1}=\left(\begin{array}{cc}i& -i\\ 0& -i\end{array}\right).$$

$$\mathrm{tr}{E}_1\in \{\pm 3,\pm 4,\pm 5,\dots \}\cup \{\pm i,\pm 2i,\pm 3i\dots \}.$$

$$\left(\begin{array}{cc}\sqrt{l_a}& 0\\ 0& 1/\sqrt{l_a}\end{array}\right)\text{ or }\left(\begin{array}{cc}\sqrt{l_a}i& 0\\ 0& 1/\sqrt{l_a}i\end{array}\right),$$

$$\mathrm{tr}{E}_a=\pm (\sqrt{l_a}+1/\sqrt{l_a})\text{ or }\pm (\sqrt{l_a}-1/\sqrt{l_a})i.$$

$$\mathrm{tr}{E}_a\in \{\pm x|x\in \mathbb{R},x\ge 3\}\cup \{\pm yi|y\in \mathbb{R},y\ge 1\},$$

$$A_a:=P_{a}Q=\left(\begin{array}{cc}-3a/2& 1/2\\ -1/2& -1/2a\end{array}\right),B:=R=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right).$$

$$G_a$$

$$\displaystyle=\left\langle A_{a},B\>\Big{|}\>\begin{array}[]{l}(A_{a}BA_{a}^{-1}BA_{a}BA_{a}^{-1})^{2}=(A_{a}BA_{a}^{-1}B^{-1}A_{a}B)^{2}=B^{2}\\ =(A_{a}BA_{a}^{-1}B^{-1}A_{a}B\cdot B)^{2}=(B\cdot A_{a}BA_{a}^{-1}BA_{a}BA_{a}^{-1})^{3}=id\end{array}\right\rangle$$

$$=⟨A_a,B|(A_{a}B{A}_a^{-1}{B}^{-1}{A}_{a}B{)}^2=B^2=(A_{a}B{A}_a^{-1}{B}^{-1}{A}_a{)}^2=id⟩.$$

$$\tau ((A_{a}B{A}_a^{-1}{B}^{-1}{A}_{a}B{)}^2)=\tau (B^2)=\tau ((A_{a}B{A}_a^{-1}{B}^{-1}{A}_a{)}^2)=(0,0),$$

$$⟨X,Y⟩\subset G_{(0,0)}\cup G_{\tau (Y)}\ne G_{(0,0)}\cup G_{(1,0)}\cup G_{(0,1)}\cup G_{(1,1)}=G$$

$$J(G_a)=J(A_a,B)=\frac{(3a^2-1{)}^2}{4a^2}.$$

$$J(A_a,B)=\left|\frac{(3a^2+1{)}^2}{4a^2}-4\right|+1=\frac{(3a^2-1{)}^2}{4a^2}.$$

$$H_n:=⟨P,Q,R|P^2=Q^2=R^2=(PR{)}^3=(QR{)}^2=(PQ{)}^n=id⟩$$

$$\pi (⟨C,D⟩)=⟨\pi (D)⟩\ne H_n,$$

$$J(C,D)=|{\mathrm{tr}}^2{A}_a-4|+|0-2|>|{\mathrm{tr}}^2{A}_a-4|+|1-2|=J(A_a,B).$$

$$J(C,D)=|{\mathrm{tr}}^{2}C-4|+|\mathrm{tr}[C,D]-2|=|{\mathrm{tr}}^2{A}_a-4|+|\pm 1-2|\ge J(A_a,B).$$

$$J(C,D)\ge |{\mathrm{tr}}^{2}A-4|+|-2-2|>J(A_a,B).$$

$$J(C,D)=|{\mathrm{tr}}^{2}C-4|+|\mathrm{tr}[C,D]-2|\ge |{\mathrm{tr}}^2{A}_a-4|+1=J(A_a,B).$$

$$J(C,D)=|{\mathrm{tr}}^{2}C-4|+|\mathrm{tr}[C,D]-2|\ge |{\mathrm{tr}}^{2}C-4|\ge 5\ge J(A_a,B).$$

$$\psi (U)+\psi (V)=\psi (X)\psi (Y).$$

$${\psi }_{\rho }(X_{n/m})=\mathrm{tr}\rho (W_{n/m}),$$

Full text

[TABLE]

If $G$ is a non-elementary Kleinian group, then $J(G)\geq 1$. This inequality is called Jørgensen’s inequality. In this paper, we show that, for any $r\geq 1$, there exists a non-elementary Kleinian group whose Jørgensen number is equal to $r$. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jørgensen numbers from above in the diagonal slice of Schottky space.

This work was supported by JSPS KAKENHI Grant Number 26400088.

1. Introduction

Let $G$ be a two generator subgroup of $\operatorname{PSL}(2,\mathbb{C})$. Determining whether or not $G$ is discrete is an important problem in Kleinian group theory. In 1976, Jørgensen [7] showed that if $G=\langle X,Y\rangle$ is a rank two non-elementary discrete subgroup of $\operatorname{PSL}(2,\mathbb{C})$ (Kleinian group), then

[TABLE]

where $[X,Y]=XYX^{-1}Y^{-1}$. The Jørgensen number of an ordered pair $(X,Y)$ is defined as

[TABLE]

and the Jørgensen number of a rank two non-elementary Kleinian group $G$ is defined as

[TABLE]

By (1), we have $J(G)\geq 1$. We can think of Jørgensen number as measuring how far from being indiscrete. Jørgensen’s inequality is sharp, and if $J(G)=1$, $G$ is called a Jørgensen group, There are many results on Jørgensen groups in the literature [5, 8, 10, 11, 12, 15, 18, 19, 23]. Also, calculating Jørgensen numbers can be difficult and interesting challenge [3, 6, 20]. Oichi-Sato [16] asked the following natural question:

Question 1.1** (The realization problem).**

Let $r$ be a real number with $r\geq 1$. When is there a non-elementary Kleinian group whose Jorgensen number is equal to $r$?

Oichi-Sato [16] claimed that if $r=1,2,3$ or $r\geq 4$, then there is a non-elementary Kleinian group whose Jørgensen number is equal to $r$. (See also [3, 23, 24].) Though Jørgensen’s inequality is sharp, constructing a Kleinian group with small Jørgensen number (in particular, less than $4$) is not easy. To the best of our knowledge, there is no known example of a classical Schottky group whose Jørgensen number is less than $4$. In this paper, we solve this realization problem.

Theorem 1.2**.**

For any real number $r\geq 1$, there is a rank two non-elementary discrete subgroup of $\operatorname{PSL}(2,\mathbb{C})$ whose Jørgensen number is $r$.

The Jørgensen numbers can be studied by computer experiments. We will present a computer generated picture of the estimates of Jørgensen numbers from above in the diagonal slice of Schottky space. At the end of the paper, we present some Jørgensen groups, and we conjecture that they are counter examples of Li-Oichi-Sato’s conjecture.

In section 2, we give a proof of Theorem 1.2 for the case $1\leq r\leq 4$. In section 3, we give a proof for the case $4\leq r$. In section 4, we present our computer generated picture, and describe our calculation.

2. The proof for the case $1\leq r\leq 4$

In this section, we give a proof for the case $1\leq r\leq 4$. For simplicity, we use the same notation for a matrix in $\operatorname{SL}(2,\mathbb{C})$ and its equivalence class in $\operatorname{PSL}(2,\mathbb{C})$.

2.1. The basic configuration

Let $z,z^{\prime}\in\mathbb{C}$, and $[z,z^{\prime}]$ be the oriented line from $z$ to $z^{\prime}$ in $\mathbb{H}^{3}$. Following [4], we define the line matrix associated with $[z,z^{\prime}]$ as

[TABLE]

See [4], p. 64, equation (1). It is the order two rotation about $[z,z^{\prime}]$.

For $a\geq 1$, set

[TABLE]

Let $G_{a}$ be a subgroup of $\operatorname{PSL}(2,\mathbb{C})$ generated by $P_{a},Q$ and $R$. The axes of the generators are contained in the vertical plane $\{(x,0,z)|z>0\}\cong\mathbb{H}^{2}$. Since we assumed that $a\geq 1$, $[a,-3a]$ and $[1,-1]$ do not intersect in $\mathbb{H}^{3}$. The angle between $[1,-1]$ and $[0,\infty]$ is $\pi/2$, and the angle between $[0,\infty]$ and $[a,-3a]$ is $\pi/3$.

See Figure 1. Hence, as an abstract group, $G_{a}$ has a presentation

[TABLE]

for any $a\geq 1$ and is isomorphic to a hyperbolic (full) triangle group in $\mathbb{H}^{2}$. (Note that hyperbolic (full) triangle groups are defined as groups generated by reflections, but $G_{a}$ is generated by rotations in $\mathbb{H}^{3}$.) Hence, $G_{a}$ is non-elementary and discrete. For later purpose, we classify non-trivial elements of $G_{a}$ into three types:

1. (i)

elliptic elements 2. (ii)

loxodromic elements which can be conjugated into $\langle P_{a},Q\rangle$ 3. (iii)

loxodromic elements which can not be conjugated into $\langle P_{a},Q\rangle$

When $a=1$, we define that the parabolic elements are of type (ii).

Remark 2.1*.*

Groups similar to $\langle P_{a},Q,R\rangle$ were studied by C. Series, S. P. Tan and the first author in [21].

Lemma 2.2**.**

Let $E_{a}$ be an element of type (iii) in $G_{a}$. Then, we have

[TABLE]

Proof.

We begin by considering the case $a=1$. Set

[TABLE]

Then, we have

[TABLE]

Hence, for any element $N\in G_{1}$, $\operatorname{tr}N=\pm n$ or $\pm n\>i$ for some integer $n$. Let $E_{1}$ be an element of type (iii) in $G_{1}$. Since $E_{1}$ is loxodromic,

[TABLE]

Now, we consider the general case. For $a>1$, let $E_{a}$ be a type (iii) element of $G_{a}$. Since $E_{a}$ is loxodromic and preserves the vertical plane $\{(x,0,z)|x>0\}\cong\mathbb{H}^{2}$, $E_{a}$ or $E_{a}^{-1}$ is conjugate to

[TABLE]

where $l_{a}$ is the translation length of $E_{a}$. Thus, we have

[TABLE]

Note that the region $F_{a}$ in the vertical plane $\mathbb{H}^{2}$ bounded by the axes of $P_{a}$, $Q$, $R$ is a fundamental region for $G_{a}$ in $\mathbb{H}^{2}$. Since $F_{a}$ is monotonically increasing in $a$, $l_{a}$ is also monotonically increasing in $a$. Combining (2), we see that

[TABLE]

and the lemma is proved. ∎

Remark 2.3*.*

By this proof, we see that $G_{1}$ is a conjugate of a subgroup of Picard group. Sato [20] showed that Picard group is a Jorgensen group. González-Acuña and Ramírez [6] described all Jørgensen subgroups of Picard group.

2.2. The singular solid torus

We define

[TABLE]

Since $Q=A_{a}BA_{a}^{-1}B^{-1}A_{a}B$, the rank of $G_{a}$ is two and $(A_{a},B)$ is a generating pair of $G_{a}$. As an abstract group, by Tietze transformation, we have

[TABLE]

Remark 2.4*.*

Generating pairs similar to $A_{a}$ and $B$ were studied in [21], and called the singular solid torus.

Lemma 2.5**.**

In $G_{a}$, any element of order three can not be a part of minimal generating system.

Proof.

Let $\tau$ be the natural projection from the free group $F_{2}=\langle A_{a},B\rangle$ of rank two to $\langle A_{a}\>|\>A_{a}^{2}\rangle\oplus\langle B\>|\>B^{2}\rangle\cong\{(i,j)\>|\>i,j\in\{0,1\}\}$. Since

[TABLE]

$\tau$ is well-defined on $G_{a}$. We denote the set $\{X\in G_{a}\>|\>\tau(X)=(i,j)\}$ by $G_{(i,j)}$. $G_{(i,j)}$ is not empty set for $(i,j)=(0,0),(0,1),(1,0),(1,1)$.

Let $X$ be an element of $G$ such that $X^{3}=id$. Then, $\tau(X)=(0,0)$. (Otherwise, we have $\tau(X^{3})\equiv\tau(X^{2})+\tau(X)\equiv\tau(X)\not\equiv(0,0)=\tau(id)$ (mod 2), which is a contradiction.) Hence, for any $Y\in G$, we have

[TABLE]

and the lemma is proved. ∎

2.3. The Jørgensen number

Finally, we calculate the Jørgensen number.

Proposition 2.6**.**

For any $1\leq a\leq a_{0}$, where $a_{0}=(\sqrt{7}+2)/3$, we have

[TABLE]

In particular, $J(G_{1})=1$ and $J(G_{a_{0}})=4$. For any $1\leq r\leq 4$, the Jørgensen number is realized.

Proof.

Since $\operatorname{tr}^{2}A=(3a^{2}+1)^{2}/4a^{2}$ and $\operatorname{tr}[A_{a},B]=1$, for $a\geq 1$, we have

[TABLE]

Since $a_{0}$ satisfies the equation $(3a_{0}^{2}-1)^{2}/4a_{0}^{2}=4$, if $1\leq a\leq a_{0}$, we have $1\leq J(A_{a},B)\leq 4$. To prove this proposition by contradiction, suppose that there exist $C,D\in G_{a}$ such that $J(C,D)<J(A_{a},B)$ and $G_{a}=\langle C,D\rangle$. Since $G_{a}$ is non-elementary, $|\operatorname{tr}[C,D]-2|>0$. (See p.68 in [1].)

Case 1 $C$ is of type (i). If $C$ is an elliptic element of order two, then $\operatorname{tr}C=0$ and $J(C,D)=|0^{2}-4|+|\operatorname{tr}[C,D]-2|>4\geq J(A_{a},B)$, and this can not happen. If $C$ is an elliptic element of order three, since $C$ can not be a part of minimal generating system by Lemma 2.5, this can not happen.

Case 2 $C$ is of type (ii). Let $C^{\prime}$ be an element of $\langle P_{a},Q\rangle$ which is conjugate to $C$. Since $\langle P_{a},Q\rangle$ is isomorphic to the infinite dihedral group, $C^{\prime}$ can be written as $C^{\prime}=P_{a}QP_{a}Q\cdots$ or $C^{\prime}=QP_{a}QP_{a}\cdots$. If the word length is odd, then $C^{\prime}$ is conjugate to $P_{a}$ or $Q$ and $C^{\prime}$ is not loxodromic. Thus, the word length is even, and $C^{\prime}=(P_{a}Q)^{n}$ or $C^{\prime}=(QP_{a})^{-n}=(P_{a}Q)^{n}$ for some $n\in\mathbb{Z}$.

Case 2-1 $|n|>1$. We claim that $G_{a}\neq\langle C,D\rangle$. To see this, consider the Coxeter group

[TABLE]

and let $\pi:G_{a}\to H_{n}$ be the natural surjection. Since $\pi(C)=id$, we have

[TABLE]

because $H_{n}$ is not cyclic when $|n|>1$. It follows that $G_{a}\neq\langle C,D\rangle$ and the claim is proved.

Case 2-2 $|n|=1$. In this case, $\operatorname{tr}^{2}C=\operatorname{tr}^{2}A_{a}$.

Case 2-2-1 $[C,D]$ is of type (i). If $[C,D]$ is elliptic of order two, then

[TABLE]

If $[C,D]$ is elliptic of order three, then $\operatorname{tr}[C,D]=\pm 1$, and

[TABLE]

Case 2-2-2 $[C,D]$ is of type (ii). Let $E_{a}^{\prime}$ be an element of $\langle P_{a},Q\rangle$ which is conjugate to $[C,D]$. Then, $E_{a}^{\prime}=A_{a}^{m}$ for some $m\neq 0$. (Recall the first paragraph of Case 2.) We have $\lim_{a\to 1}\operatorname{tr}E_{a}^{\prime}=\lim_{a\to a}\operatorname{tr}A_{a}^{m}=2$ or $-2$. For $1\leq b\leq a$, we denote the elements in $G_{b}$ which corresponds to $C$ and $D$ by $C_{b}$ and $D_{b}$.

If $\lim_{a\to 1}\operatorname{tr}E_{a}^{\prime}=2$, then $J(C_{b},D_{b})$ is close to [math] when $b$ is close to $1$, which contradicts Jørgensen’s inequality, because $G_{b}$ is non-elementary and discrete.

If $\lim_{a\to 1}\operatorname{tr}E_{a}^{\prime}=-2$, then $\operatorname{tr}[C,D]\leq-2$ and

[TABLE]

Case 2-2-3 $[C,D]$ is of type (iii). By Lemma 2.2, we have

[TABLE]

Case 3 $C$ is of type (iii). By lemma 2.2, we have

[TABLE]

Hence, $J(G_{a})=J(A_{a},B)$ for $1\leq a\leq a_{0}$, and the proposition is proved. ∎

3. The proof for the case $r\geq 4$

Oichi and Sato [16] claimed that, for every real number $r\geq 4$, there is a subgroup $G$ of $\operatorname{PSL}(2,\mathbb{C})$ such that $J(G)=r$. But the proof was not written. For completeness, we give a proof of this fact by calculating the Jørgensen number of some Kleinian groups.

3.1. Markoff maps

First, we recall Bowditch, Tan-Wong-Zhang theory [2, 22] on Markoff maps very briefly. See section 3 in [22] for detail.

We denote the set $\mathbb{Q}\cup\{1/0\}$ by $\widehat{\mathbb{Q}}$. Let $\mathcal{F}$ be the Farey triangulation of the upper half plane $\mathbb{H}^{2}$. Recall that the vertex set of $\mathcal{F}$ is $\widehat{\mathbb{Q}}$, and two vertices $p/q$ and $r/s$ are connected by a geodesic in $\mathbb{H}^{2}$ if $ps-qr=\pm 1$. Let $\Sigma$ be the binary tree dual to $\mathcal{F}$. See Figure 2.

A complementary region of $\Sigma$ is the closure of a connected component of the complement of $\Sigma$. The set of complementary regions of $\Sigma$ is denoted by $\Omega$. Since each complementary region corresponds to a vertex in $\mathcal{F}$, we can identify $\Omega$ with $\widehat{\mathbb{Q}}$, and we denote the complementary region which corresponds to $p/q\in\mathbb{Q}$ by $X_{p/q}$. Let $e$ be an edge of $\Sigma$ with end points $u$ and $v$. Then, there exist $X,Y,U,V\in\Omega$ such that $e=X\cap Y$, $u=X\cap Y\cap U$ and $v=X\cap Y\cap V$. See Figure  3.

We write $e=(X,Y;U,V)$ to indicate these regions. A Markoff map is a map $\psi$ from $\Omega$ to $\mathbb{C}$ such that, for every edge $e=(X,Y;U,V)$ in $\Sigma$, we have

[TABLE]

This condition is called the edge relation. Given $k\geq 0$, the set $\Omega_{\psi}(k)$ is defined by $\Omega_{\psi}(k)=\{X\in\Omega\>|\>|\psi(X)|\leq k\}$. We will need the following lemma.

Lemma 3.1** (Theorem 3.1 (2), [22]).**

Let $\psi$ be a Markoff map. For any $k\geq 2$, the union $\cup_{X\in\Omega_{\psi}(k)}X$ is connected as a subset of $\mathbb{H}^{2}$.

Let $F_{2}=\langle A,B\rangle$ be the free group on $A$ and $B$. An element $W$ of $F_{2}$ is called primitive if there exists an element $V$ such that $F_{2}=\langle W,V\rangle$, and $W$ and $V$ are called associated primitives. Let $\alpha$ be the abelianization homomorphism from $F_{2}$ onto $F_{2}/F_{2}^{\prime}$ ($\cong\mathbb{Z}^{2}$). Let $\rho$ be an $\operatorname{SL}(2,\mathbb{C})$ representation of $F_{2}$. Then, the next map $\psi_{\rho}:\Omega\to\mathbb{C}$ is well-defined:

[TABLE]

where $W_{n/m}$ is a primitive element such that $\alpha(W_{n/m})=A^{m}B^{n}$. See Corollary 3.2, [17]. For example, we have

[TABLE]

By Theorem 1.2 and 1.3 in [17] and the trace identity in $\operatorname{SL}(2,\mathbb{C})$

[TABLE]

we have the next lemma. (See Section 3, Natural correspondence ${\mathcal{X}}_{\mu-2}=\psi_{\mu}$ [22].)

Lemma 3.2**.**

$\psi_{\rho}$* is a Markoff map.*

For any associated primitives $\{A,B\}$ and $\{C,D\}$ of $F_{2}$, the commutator $[C,D]$ is conjugate to $[A,B]$ or $[A,B]^{-1}$. (See Theorem 3.9 in [13].) Thus, we have $\operatorname{tr}\rho[A,B]=\operatorname{tr}\rho[C,D]$. It follows that

Lemma 3.3**.**

If $G$ is a subgroup of $\operatorname{PSL}(2,\mathbb{C})$ isomorphic to $F_{2}$, then the trace of the commutator of associated primitives of $G$ does not depend on the choice of associated primitives.

Recall that, if $G$ is in the Maskit slice [9, 14], then it is discrete and isomorphic to $F_{2}$ and can be normalized as $G_{\mu}=\langle A,B_{\mu}\rangle$, where

[TABLE]

Corollary 3.4**.**

If $G_{\mu}$ is in the Maskit slice, then $J(G_{\mu})=4$.

Proof.

Since $|\operatorname{tr}^{2}A-4|=0$, by Lemma 3.3, we have $J(G_{\mu})=J(A,B_{\mu})=4$.. ∎

3.2. Kissing Schottky groups

We consider kissing Schottky groups studied in [14]. For a positive real number $k$, let $\rho_{k}$ be the $\operatorname{SL}(2,\mathbb{C})$ representation of $F_{2}=\langle A,B\rangle$ given by

[TABLE]

where $x,y$ are positive real numbers with $x^{2}=y^{2}+1$ and $y^{2}=2/(k+1/k)$. (See chapter 6, p.170 [14].) The first condition guarantees that $A$ and $B$ have determinant $1$. The second condition guarantees that $\operatorname{tr}[A,B]=-2$. See Figure 6.8 in [14]. For the sake of simplicity, we denote the Markoff map $\psi_{\rho_{k}}$ of the representation $\rho_{k}$ by $\psi$.

Lemma 3.5**.**

We have $\Omega_{\psi}(2x)=\{X_{0/1},X_{1/0}\}$.

Proof.

Since $\psi(X_{0/1})=\operatorname{tr}\rho_{k}(A)=2x$ and $\psi(X_{1/0})=\operatorname{tr}\rho_{k}(B)=2x$, by definition, we have $\{X_{0/1},X_{1/0}\}\subset\Omega_{\psi}(2x)$.

Complementary regions which meet $X_{1/0}$ are

[TABLE]

See Figure 2. Put $x_{n}=\psi(X_{n/1})$ for $n\in\mathbb{Z}$. Then, we have

[TABLE]

By the edge relation, we have

[TABLE]

for any $n\in\mathbb{Z}$. Observe that $|x_{n+1}|>|x_{n}|$ for any $n\geq 0$. First, direct calculation shows that $|x_{1}|>|x_{0}|$. Then, since $2(k+1)/\sqrt{k^{2}+1}>2$, the edge relation implies that $|x_{n+1}|>|x_{n}|$. We also see that $|x_{n-1}|>|x_{n}|$ for any $n\leq 0$. (In fact, $|x_{n}|=|x_{-n}|$.) Hence, $X_{0/1}$ is the only region in $\Omega_{\psi}(2x)$ which meets $1/0$.

Complementary regions which meet $X_{0/1}$ are

[TABLE]

By the edge relation, we have $\psi(X_{1/n})=\psi(X_{n/1})$. Hence, $X_{1/0}$ is the only region in $\Omega_{\psi}(2x)$ which meets $X_{0/1}$,

Since $2x=2(k+1)/\sqrt{k^{2}+1}>2$, by Lemma 3.1, $\Omega_{\psi}(2x)$ is connected. Hence, $\Omega_{\psi}(2x)=\{X_{0/1},X_{1/0}\}$, and the lemma is proved. ∎

By this lemma, we have $|\operatorname{tr}\rho_{k}(C)|\geq|\operatorname{tr}\rho_{k}(A)|$ for any primitive element $C\in F_{2}$. Since $\operatorname{tr}\rho_{k}(A)$ is real and greater than $2$, we have $|\operatorname{tr}^{2}\rho_{k}(C)-4|\geq|\operatorname{tr}^{2}\rho_{k}(A)-4|$. Since $\rho_{k}(F_{2})$ is isomorphic to $F_{2}$, by Lemma 3.3, we have the following.

Proposition 3.6**.**

For any kissing Schottky group (representation) $\rho_{k}$, we have

[TABLE]

In particular, we have $J(\rho_{1}(F_{2}))=8$, and $\lim_{k\to\infty}J(\rho_{k}(F_{2}))=4$, For any real $r$ with $4<r\leq 8$, the Jørgensen number is realized by a kissing Schottky group.

3.3. $\theta$-Schottky groups

Here, we consider Fuchsian Schottky groups described in Project 4.2, p.118 in [14]. For $0<\theta\leq\pi/4$, let $\rho_{\theta}$ be the $\operatorname{SL}(2,\mathbb{C})$ representation of $F_{2}=\langle A,B\rangle$ given by

[TABLE]

See Figure 4.10 in [14]. We denote the Markoff map $\psi_{\rho_{\theta}}$ by $\psi$.

Lemma 3.7**.**

We have $\Omega_{\psi}(2/\sin\theta)=\{X_{0/1},X_{1/0}\}$.

Proof.

Note that

[TABLE]

Then, the rest of the proof is the same as for Lemma 3.5. ∎

Hence, $|\operatorname{tr}^{2}\rho_{\theta}(C)-4|\geq|\operatorname{tr}^{2}\rho_{\theta}(A)-4|$ for any primitive $C\in F_{2}$. Since $\rho_{\theta}(F_{2})$ is isomorphic to $F_{2}$, by Lemma 3.3:

Proposition 3.8**.**

For any $\theta$-Schottky group (representation) $\rho_{\theta}$, we have

[TABLE]

In particular, we have $J(\rho_{\pi/4}(F_{2}))=8$, and $\lim_{\theta\to 0}J(\rho_{\theta}(F_{2}))=\infty$, For any real $r$ with $r\geq 8$, the Jørgensen number is realized by a $\theta$-Schottky group.

Remark 3.9*.*

$\rho_{\pi/4}$ ($\theta=\pi/4$) in this subsection and $\rho_{1}$ ($k=1$) in the last subsection are the same representation.

4. The diagonal slice of Schottky space

Jørgensen numbers can be studied by computer experiments. In this section, we present a computer generated picture (Figure 4) which estimates Jørgensen numbers in the diagonal slice of Schottky space.

4.1. The diagonal slice of Schottky space

Let us go back to the singular solid torus in section 2. Recall that

[TABLE]

In this section, we consider that $A_{a},B\in\operatorname{SL}(2,\mathbb{C})$. We denote $\langle A_{a},B\rangle$ by $G_{a}$. Set

[TABLE]

Then, by the trace identity (6), we have

[TABLE]

See section 5.0.2 (in particular, Remark 5.4 for the sign of the square roots) in [21]. From now on, we consider that $x$ is a complex parameter. The locus $\mathcal{D}$ in the $x$-plane of discrete and faithful representations was fully determined by computing Keen-Series pleating rays [21]. (Here, “faithful” means that $\langle A_{a},B\rangle$ is isomorphic to $\langle A_{1},B\rangle$ as an abstract group.) $\mathcal{D}$ is called the diagonal slice of Schottky space. See Figure 4.

The outside of the center black eye corresponds to $\mathcal{D}$. $\mathcal{D}$ is foliated by pleating rays. We will describe pleating rays briefly in the next subsection.

Let $a$ be a complex number such that $x=-\operatorname{tr}A_{a}^{2}\in\mathcal{D}$. We denote the representation from $F_{2}=\langle{\mathcal{A}},{\mathcal{B}}\rangle$ to $\operatorname{SL}(2,\mathbb{C})$ which sends $\mathcal{A}$ to $A_{a}$ and $\mathcal{B}$ to $B$ by $\rho_{x}$. Let $\psi_{x}:\widehat{\mathbb{Q}}\to\mathbb{C}$ be the Markoff map associated with $\langle A_{a},B\rangle$:

[TABLE]

If $W$ and $V$ are associated primitives of $F_{2}$, then their images in $G_{a}$ generate $G_{a}$. By Lemma 3.3 and equation (9), $\operatorname{tr}\rho_{x}([W,V])=\operatorname{tr}[A_{a},B]=1$. Hence,

[TABLE]

where $n/m$ is determined by the abelianization $\alpha(W)={\mathcal{A}}^{m}{\mathcal{B}}^{n}$. Hence,

[TABLE]

gives an estimate of $J(G_{a})$ from above. We calculate $\Psi(x)$ by computer for $\{x\in\mathbb{C}\>|\>-7\leq\Re(x)\leq 8,-5\leq\Im(x)\leq 5\}$. The color outside the black eye in Figure 4 indicates the values of $\Psi(x)$. Since thees groups are discrete and faithful (in particular non-elementary), $\Psi(x)\geq 1$. Since $|\psi^{2}_{x}(1/0)-4|+1=5$, we have $\Psi(x)\leq 5$. Figure 4 suggests that for each $1\leq r\leq 5$, there are many non-elementary Kleinian groups in the diagonal slice with Jørgensen number $r$.

Remark 4.1*.*

In practice, we can not calculate $\psi_{x}^{2}(q)$ for all $q\in\widehat{\mathbb{Q}}$, and Figure 4 is an approximation of $\Psi(x)$ by calculating $\psi_{x}^{2}(q)$ for many $q\in\widehat{\mathbb{Q}}$. But, if $x$ is in the Bowditch set, in principle, we can calculate $\Psi(x)$. The key is Lemma 3.24 in [22]. The diagonal slice of Schottky space seems to coincide with the Bowditch set. See Section 2, [21].

Question 4.2**.**

Does the equation $J(G_{a})=\Psi(x)$ hold for each $x\in\mathcal{D}$? Note that, since $G_{a}$ is not free, there might be generating pairs of $G_{a}$ which do not come from associated primitives in $F_{2}$.

4.2. Jørgensen groups on the boundary of the diagonal slice

We consider the case $\Psi(x)=1$ on $\partial\mathcal{D}$. We begin by describing pleating rays very briefly. See [21] for detail.

For $p/q\in\widehat{\mathbb{Q}}$, the real trace locus $\mathbb{R}_{p/q}$ of $p/q$ is $\{x\in\mathbb{C}\>|\>\psi_{x}(p/q)\in(-\infty,-2]\cup[2,\infty)\}$. The rational pleating ray ${\mathcal{P}}_{p/q}$ is a union of connected non-singular branches of $\mathbb{R}_{p/q}$ (Corollary 4.11 [21]). The rational pleating rays are indexed by $\widehat{\mathbb{Q}}/\sim$, where $p/q\sim p^{\prime}/q^{\prime}$ if and only if $p^{\prime}/q^{\prime}=\pm p/q+2k,k\in\mathbb{Z}$ (Proposition 4.8, [21]). (If $p/q\sim p^{\prime}/q^{\prime}$, then we have $\psi_{x}(p/q)=\pm\psi_{x}(p^{\prime}/q^{\prime})$.) For example, the condition $a\in\mathbb{R}$, $1\leq a\leq(\sqrt{7}+2)/3$ in subsection 2.3 corresponds to $\{x\in\mathbb{R}\>|\>-5\leq x\leq-2\}\subset{\mathcal{P}}_{0/1}$.

If $p/q$ is not an integer, ${\mathcal{P}}_{p/q}$ consists of two components (branches). One is in the upper-half plane, and the other in the lower-half plane. Let $e_{p/q}$ be the end point of ${\mathcal{P}}_{p/q}$ in the upper-half plane, or on the real axis if $p/q$ is an integer. By construction, we have $\psi_{e_{p/q}}^{2}(p/q)-4=0$. Hence, $\Psi(e_{p/q})=1$, and the corresponding representation is a Jørgensen group of parabolic type.

In order to describe these groups, we recall the Li-Oichi-Sato normalization:

Lemma 4.3** (Lemma 3.1 [11]).**

Let $M$ and $N$ be elements of $\operatorname{PSL}(2,\mathbb{C})$ such that $M$ is parabolic and $N$ is elliptic or loxodromic. Then, $M,N$ can be normalized as

[TABLE]

where $\sigma\in\mathbb{C}\setminus\{0\}$ and $\mu\in\mathbb{C}$.

Remark 4.4*.*

Li-Oichi-Sato [11] [10] [12] considered the case $\mu=ik$, $k\in\mathbb{R}$. They conjectured that, for any Jørgensen group $G$ of parabolic type, there exists a marked group $G_{\sigma,ik}=\langle M,N_{\sigma,ik}\rangle$, $\sigma\in\mathbb{C}\setminus\{0\}$, $k\in\mathbb{R}$, such that $G_{\sigma,ik}$ is conjugate to $G$. Later, Callahan [3] found counter examples for this conjecture.

Now, we compute some examples. We begin by calculating some Markoff maps and the end point of pleating rays:

[TABLE]

(Note that the equation $\psi_{x}^{2}(p/q)=4$ have many roots, and in order to make the right choice for $e_{p/q}$, we need [21].) Let $A_{p/q}$ denote the matrix in (7) such that $e_{p/q}=-\operatorname{tr}A_{a}^{2}$. Then, for example,

[TABLE]

generate Jørgensen groups of parabolic type. The Li-Oichi-Sato parameters for these groups are as follows:

[TABLE]

We conjecture that groups which correspond to the end points of the rational pleating rays (except $-2$ and $3$ on the real axis) are counter examples for Li-Oichi-Sato’s conjecture in Remark 4.4.

Remark 4.5*.*

Let $y$ be an element of $\mathcal{D}$ such that $\psi_{y}$ has a sequence $\{q_{n}\}_{n\in\mathbb{Z}}$ of distinct elements of $\widehat{\mathbb{Q}}/\sim$ with $\lim_{n\to\infty}\psi_{y}^{2}(q_{n})=4$. Then $\Psi(y)=1$ and $y$ corresponds to a geometrically infinite group with unbounded geometry.

Remark 4.6*.*

The limit set of $\langle M,N_{\sigma,\mu}\rangle$ with $(\sigma,\mu)=(i,0)$ is the real line. The limit sets of other examples are complicated and beautiful. For example, see Figure 5 for the limit set of the third case $(\sigma,\mu)\approx(i,0.1839+0.9356i)$.